where W_0 is the initial wealth, W_t is the wealth after t years, r is the nominal annual interest rate, and n is the number of compounding periods per year.

As we vary the number of compounding periods, we notice that we get a higher return as the number goes up. Note that this is the case even though we divide the interest rate r by the number of compounding periods n.

The effect of compound interest, with an initial investment of $1,000 and 20% annual interest, compounded at various frequencies. Source. Licensed under CC-BY-SA-3.0

Since we’ve noticed that we get more money as we increase the number of compounding periods, we might be interested in what happens when we grant ourselves infinitely many compounding periods. Contrary to what you might think, we don’t get infinite money.

Since we know that \left(1 + \frac{1}{n}\right)^n converges to e, we get

W_t = W_0\,e^{rt}

This corresponds to the purple line in the chart above. As you can see, it results in the highest growth for the given interest rate.

Now, we might be interested in the rate, which when compounded continuously, would result in the same growth as annual compounding, but smoothly distributed throughout the year. In the graph, this would correspond to a continuous line that touches every left corner of the annual line (green).

The continuously compounded rate r_c for this curve would be slightly lower than the corresponding annual interest rate r_a. But by how much?

We already have the formulas for discrete and periodic compounding, so we can equate them and solve for r_c: